Difference between revisions of "Time scale discrete Fourier transform"

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(Created page with "Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a frequency root of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is def...")
 
 
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Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a [[frequency root]] of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is defined by
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Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a [[frequency roots|frequency root]] of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is defined by
 
$$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$
 
$$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$
 
where $e_{z_n}$ denotes the [[delta exponential]].
 
where $e_{z_n}$ denotes the [[delta exponential]].

Latest revision as of 04:00, 26 February 2018

Let $\mathbb{T}=\{t_0,t_1,\ldots,t_n\}$ be a finite time scale and let $z_n$ be a frequency root of $\mathbb{T}$. The time scale discrete Fourier transform at $z_k$ is defined by $$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$ where $e_{z_n}$ denotes the delta exponential.

References